France tested several times in 1995 and 1996, many, including that January 1996 one, were related to the development of the TN-75 warhead. This particular warhead was one of the last warheads France deployed before stopping testing, and was key to their new, post-Cold War status as a submarine-only nuclear power (France and the UK both only use submarine-launched ballistic missiles as their delivery platforms).

As for why they felt they needed to test it repeatedly, at various yields, I don't know and I'm not sure they've said (all nuclear powers can be reticent in talking about weapons currently deployed, and the French are especially secretive with regards to their nuclear weapons work).

But just speaking generally, SLBM warheads, especially MIRVed ones (multiple warheads per missile; each M45 SLBM carries 6 TN-75 warheads), have very constrained volume and mass profiles. They need to be both physically pretty small and not too heavy. That means that they typically push the envelope in terms of weapons design: they are ultra-compact, utilize many clever engineering tricks, and seem to use exotic and unusual materials. They are trying to squeeze every little ounce of energy out of their nuclear material, with not much room for error. This kind of design traditionally has taken the other nuclear powers many tests to "validate," to have confidence that it will work when and how you desire it to, because it is not "conservative" in the slightest.

It is a major question in the US nuclear community right now as to whether the US can continue to not test its nuclear warheads, and the amount of uncertainty that grows every decade that goes by. It is hard for me to imagine the US not doing some kind of testing if it tried to deploy new warhead designs — the confidence in an "on paper" design is not so high, even though historically the US got very good at predicting performance of designs from first principles. On the confidence issue, Donald MacKenzie's Inventing Accuracy is a very good discussion of this, from the context of ICBM development.

As for whether they achieved what they intended, I mean, we don't really know what the specific questions they were asking of the tests. But they did deploy the TN-75 so it seems they got something useful out of them. Or one would hope.